See National Math Standards for this Challenge.
In one way this is a bit of a trick question. Since both the comet and the impactor spacecraft will be moving in nearly the same orbit at the time of impact, what is important really is how fast the impactor spacecraft will be traveling relative to the comet. As an example of the importance of this difference, consider two cars involved in a rear-end collision. If both cars are moving at the time they collide, it isn't the velocity of the first car or the second car that would be used to calculate the energy of impact, it's the difference between them. So a car traveling at 60 mph striking a car traveling 55 mph transfers the same amount of energy at collision as a car traveling 5 mph striking a stationary car.
Another complication that has to be taken into account is the fact that the comet and impactor spacecraft will be moving through three-dimensional space, instead of the "one dimensional" space of the car example. Fortunately this can be fixed by thinking of the motion one piece at a time, using vectors.
If you assign the same x-y-z set of axes for both objects, their movement can be broken down into three components, an x-component, a y-component and a z-component. This gives you the velocity that the object is moving along that axis. The overall velocity of the object can be determined by simply adding up the vectors. Since the three vectors will be at right angles to each other (since they're along the three right-angle axes), we can just use the three dimensional form of the Pythagorean theorem:
Looking back at the car example, let's say that the cars were moving along the x-axis. This means that their velocities (in mph) along the three axes would be:
Car 1: (60, 0, 0)
Car 2: (55, 0, 0)
This means that the overall velocities of the cars would be:
Car 1: = 60 mph
Car 2: = 55 mph
Okay, enough hints. The last piece of information you need is the x, y, and z velocities (in kilometers per second) for the comet (27.09, -11.40, -5.46) and the impactor spacecraft (18.41, -12.67, -0.16). From these, can you determine the velocity of the impactor spacecraft relative to the comet at the time of impact?